Some nice theorem to find the value of serie

Question

I know a lot of techniques to show that a serie converge or diverge (Cauchy condensation test, Abel's test, comparision with an integrale, root test, ratio test...).

On the other hand I know nothing about caculating the value of a serie. Is there any theorems that may help ?

From now on I know the following :

• integrale test can be combine with squeeze theorem
• geometric serie

For example in this [How to prove $\lim \limits_{x \to 1^-} \sum\limits_{n=0}^\infty (-1)^nx^{n²} = \frac{1}{2} \ $? there are talking about :

• Herglotz' trick

• Weierstrass products.

• ...

So I would like to know if there are some other nice theorem or slick techniques to find the value of a serie.

Thank you !

Answer 1

In this regard, series are much like integrals: you need knowledge and, even more, ingenuity and experience.

Some basic Theorems, besides those you cited, are:
- Indefinite Sum;
- Summation by parts; - (Discrete) Convolution;
- Partial Fractions Decomposition;
- Binomial Tranform and others;
- Gosper's Algorithm and Hypergeometric Function;
- Abel's Binomial Theorem
and many others

Answer 2

I don't know if this is a good approach and if you mentioned this but it would add another possible approach which I posted on the linked question.

Consider

$$ F(y+1)-F(y) = e^{i\pi y} x^{y^2} = e^{i \pi y -ay^2} = f, $$

Take fourier transform of this and solve e.g. $$ \hat F(s) = \frac{\hat f(s)}{e^{is} - 1} $$ Note that $$ \lim_{n \to \infty} \sum_{i=0}^{n} F(i+1)-F(i) = - F(0) = \lim_{n \to \infty} F(n+1) - c \int \mathcal{F}(f)(s)\,ds = - c \int \mathcal{F}(f)(s)\,ds, $$ under the assumption $\lim_{n\to\infty} F(n+1) = 0$

For the example above it turns out the fourier transform turns into a delta measure as we approach $x\to 1$ and this might be an approach that is tractable and lead to convergence proof and limit calculation.